比尔·盖茨最新书单：怎样用数学思维约到女神？
2016-05-20 17:55:32 来源: 澎湃新闻网(上海)
（原标题：比尔·盖茨最新书单，有本书教你怎样用数学思维约到女神）
【编者按】
一如既往的，比尔·盖茨在初夏更新了个人博客文章，开出了一份推荐阅读书单。
比尔·盖茨最新书单：怎样用数学思维约到女神？
比尔·盖茨最新书单中涵盖了科学、数学和小说类别的5本书籍：美国科幻作家尼尔·斯蒂芬森（Neal Stephenson）的《七夏娃》（Seveneves）、英国生物化学家尼克·莱恩（Nick Lane）的《至关重要的问题》（The Vital Question）、日本电商大佬三木谷浩史与父亲三木谷良一合著的《竞争力》、以色列历史学教授尤瓦尔·赫拉利（Noah Yuval Harari）的《人类简史：从动物到上帝》（Sapiens：A Brief History of Humankind，中文版已出），以及威斯康星大学教授乔丹·爱伦伯格（Jordan Ellenberg）所著的这本How Not to be Wrong（中文本《魔鬼数学》，中信出版社，2015年9月）。
该书旨在说明数学是一门告诉我们“如果做才不会犯错”的科学。澎湃新闻“翻书党”经授权摘编其中关于如何运用数学思维让你成功约会心仪对象的章节。
比尔·盖茨最新书单：怎样用数学思维约到女神？
威斯康星大学教授乔丹·爱伦伯格（Jordan Ellenberg）
我们先从多头绒泡菌这个有趣的微生物说起，在大部分时间里，它都表现为一种极小的单细胞形态，即，单身，但是，如果条件合适，成千上万的多头绒泡菌就会结合在一起，形成“原质团”。在这种形态下，多头绒泡菌为嫩黄色，体积也会变大，肉眼可见。在野外环境中，它生长在腐朽的植物上，而在实验室中，它最喜欢的栖息之所是燕麦。
多头绒泡菌无脑无神经，更不用说情感与思维了，但是，与所有生物一样，它也会做决策，更有意思的是，多头绒泡菌会做出非常正确的决策。当然，决策无非是关于“靠近我喜欢的东西”（燕麦）与“远离我不喜欢的东西”（明亮的光线）。研究人员在皮式培养皿的一侧放置3 克燕麦，在另一侧放置5 克燕麦并用紫外线照射燕麦，然后在培养皿的中间位置放上多头绒泡菌。他们发现，在这种情况下，多头绒泡菌选择这两个方向的次数大约各占一半，更多的食物基本抵消了紫外线给多头绒泡菌造成的不舒服的感觉。若是经济学家来分析，他们肯定会认为，对于多头绒泡菌而言，黑暗中的一小堆燕麦与明亮处的一大堆燕麦的效用是一样的，因此，多头绒泡菌会左右为难。
不过，在把5 克燕麦换成10 克之后，这种平衡完全被打破了，多头绒泡菌根本不在乎光线的问题，每次都会朝10 克燕麦靠近。这个实验告诉我们，多头绒泡菌在做决策时会优先考虑哪些因素，在这些因素相互矛盾时又是如何做出选择的。从这些实验来看，多头绒泡菌似乎相当理性。
但是，当实验者把多头绒泡菌放到皮式培养皿中之后，给了它们三种选择：在黑暗处放置3 克燕麦（3– 黑暗），在明亮处放置5 克燕麦（5– 明亮），在黑暗处放置1 克燕麦（1– 黑暗）。我们可能会认为多头绒泡菌绝不可能靠近1– 黑暗，因为3– 黑暗的燕麦数量更多，具有明显的优势。的确，多头绒泡菌几乎一次也没有选择1– 黑暗。
我们还可能会进一步猜测，既然在之前的条件下，3– 黑暗与5– 明亮对多头绒泡菌具有同样的吸引力，那么，在新的条件下，应该会继续出现这样的情况。用经济学家的话来说，新的选择方案不会改变3– 黑暗与5– 明亮效用相同的事实。但是，实验结果并非如此：在有1– 黑暗可选的情况下，多头绒泡菌的喜好发生了变化，选择3– 黑暗的次数是5– 明亮的三倍！
这是怎么回事呢？
数学领域的一个流行术语—“无关选项的独立性”（independence of irrelevant alternatives），根据这个法则，无论你是多头绒泡菌、人还是民主国家，如果要在方案甲与方案乙之间做选择，第三个方案丙的出现都不会影响你对甲和乙的倾向性。如果你在为购买丰田普锐斯还是悍马犹豫不决，福特斑马对你到底买哪款车的选择不会产生任何影响，因为你知道自己肯定不会购买福特斑马。
人也是如此，我们把燕麦换成浪漫的伴侣。我们将为你提供几个虚构的人物，请把这几个人想象成你未来的约会对象，然后从这些人中选择一个与之约会。假定这几个未来的约会对象都满足以下条件：（1）北卡罗来纳大学（或者杜克大学）的学生；（2）与你同一个民族或种族；（3）与你年龄相当。我们会描述他们的几个特点，并就每种特点给出百分位数。这些百分位数反映了他们的某种特点在相同性别、种族与年龄的北卡罗来纳大学（或者杜克大学）学生中的相对位置。
亚当的魅力处于第81 百分位数，可信度处于第51 百分位数，智力处于第65 百分位数；
比尔的魅力处于第61 百分位数、可信度处于第51 百分位数，智力处于第87 百分位数。
与多头绒泡菌一样，这些实验对象面临着艰难的选择。他们给出的答案也与多头绒泡菌一样，选择亚当和比尔作为未来约会对象的大学生各占总数的一半。
但是，在克里斯出现之后，情况发生了变化。克里斯的魅力与可信度分别处于第81 百分位数和第51 百分位数，但是他的智力与亚当一样，处于第54 百分位数。克里斯是一个“无关选项”，因为他明显逊色于亚当和比尔。结果我们应该可以猜到：在稍有逊色的新版亚当出现之后，正版亚当似乎变得更有吸引力了。当面对亚当、比尔和克里斯这3 个选项时，接近2/3 的女性选择亚当作为约会对象。
所以，如果你是一位正在寻找真爱的单身汉，那么，在考虑与哪位朋友一起去城里赴心仪对象的约会时，应该选择条件与你相似但略微逊色于你的那位。
非理性从何而来呢？我们已经知道，完全理性的个体在集体行动中有可能扭曲真实的民意。但是经验告诉我们，个人不可能是完全理性的。关于多头绒泡菌的研究表明，我们的日常行为之所以自相矛盾或不一致，可能基于某种更彻底的理由。个人之所以不理性，也许是因为他们并不是真正的个体。每个人都是一个小国家，我们要做的就是尽可能地处理各种争端、做出妥协，而最后得到的未必都是合理的结果。就像多头绒泡菌一样，我们也有可能小错不断，但却能做到大错不犯。http://tech.163.com/16/0520/17/BNHDR6PG000915BF.html
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Announcement 300: New challenges on the division by zero z/0=0\\
(2016.05.22)}
\author{{\it Institute of Reproducing Kernels}\\
Kawauchi-cho, 5-1648-16,\\
Kiryu 376-0041, Japan\\
%\date{\today}
\maketitle
{\bf Abstract: } In this announcement, for its importance we would like to state the
situation on the division by zero and propose basic new challenges.
\bigskip
\section{Introduction}
%\label{sect1}
By a {\bf natural extension} of the fractions
\begin{equation}
\frac{b}{a}
\end{equation}
for any complex numbers $a$ and $b$, we found the simple and beautiful result, for any complex number $b$
\begin{equation}
\frac{b}{0}=0,
\end{equation}
incidentally in \cite{s} by the Tikhonov regularization for the Hadamard product inversions for matrices and we discussed their properties and gave several physical interpretations on the general fractions in \cite{kmsy} for the case of real numbers.
The division by zero has a long and mysterious story over the world (see, for example, Google site with the division by zero) with its physical viewpoints since the document of zero in India on AD 628, however,
Sin-Ei Takahasi (\cite{kmsy}) established a simple and decisive interpretation (1.2) by analyzing the extensions of fractions and by showing the complete characterization for the property (1.2):
\bigskip
{\bf Proposition 1. }{\it Let F be a function from ${\bf C }\times {\bf C }$ to ${\bf C }$ satisfying
$$
F (b, a)F (c, d)= F (bc, ad)
$$
for all
$$
a, b, c, d \in {\bf C }
$$
and
$$
F (b, a) = \frac {b}{a }, \quad a, b \in {\bf C }, a \ne 0.
$$
Then, we obtain, for any $b \in {\bf C } $
$$
F (b, 0) = 0.
$$
}
Note that the complete proof of this proposition is simply given by 2 or 3 lines.
\medskip
We thus should consider, for any complex number $b$, as (1.2);
that is, for the mapping
\begin{equation}
w = \frac{1}{z},
\end{equation}
the image of $z=0$ is $w=0$ ({\bf should be defined}). This fact seems to be a curious one in connection with our well-established popular image for the point at infinity on the Riemann sphere. Therefore, the division by zero will give great impacts to complex analysis and to our ideas for the space and universe.
However, the division by zero (1.2) is now clear, indeed, for the introduction of (1.2), we have several independent approaches as in:
\medskip
1) by the generalization of the fractions by the Tikhonov regularization or by the Moore-Penrose generalized inverse,
\medskip
2) by the intuitive meaning of the fractions (division) by H. Michiwaki,
\medskip
3) by the unique extension of the fractions by S. Takahasi, as in the above,
\medskip
4) by the extension of the fundamental function $W = 1/z$ from ${\bf C} \setminus \{0\}$ into ${\bf C}$ such that $W =1/z$ is a one to one and onto mapping from $ {\bf C} \setminus \{0\} $ onto ${\bf C} \setminus \{0\}$ and the division by zero $1/0=0$ is a one to one and onto mapping extension of the function $W =1/z $ from ${\bf C}$ onto ${\bf C}$,
\medskip
and
\medskip
5) by considering the values of functions with the mean values of functions.
\medskip
Furthermore, in (\cite{msy}) we gave the results in order to show the reality of the division by zero in our world:
\medskip
\medskip
A) a field structure containing the division by zero --- the Yamada field ${\bf Y}$,
\medskip
B) by the gradient of the $y$ axis on the $(x,y)$ plane --- $\tan \frac{\pi}{2} =0$,
\medskip
C) by the reflection $W =1/\overline{z}$ of $W= z$ with respect to the unit circle with center at the origin on the complex $z$ plane --- the reflection point of zero is zero,
\medskip
and
\medskip
D) by considering rotation of a right circular cone having some very interesting
phenomenon from some practical and physical problem.
\medskip
In (\cite{mos}), many division by zero results in Euclidean spaces are given and the basic idea at the point at infinity should be changed. In (\cite{ms}), we gave beautiful geometrical interpretations of determinants from the viewpoint of the division by zero. The results show that the division by zero is our basic and elementary mathematics in our world.
\medskip
See J. A. Bergstra, Y. Hirshfeld and J. V. Tucker \cite{bht} for the relationship between fields and the division by zero, and the importance of the division by zero for computer science. It seems that the relationship of the division by zero and field structures are abstract in their paper.
Meanwhile, J. P. Barukcic and I. Barukcic (\cite{bb}) discussed recently the relation between the divisions $0/0$, $1/0$ and special relative theory of Einstein. However, their logic seems to be curious and their results contradict with ours.
Furthermore, T. S. Reis and J.A.D.W. Anderson (\cite{ra,ra2}) extend the system of the real numbers by introducing an ideal number for the division by zero $0/0$.
Meanwhile, we should refer to up-to-date information:
{\it Riemann Hypothesis Addendum - Breakthrough
Kurt Arbenz
https://www.researchgate.net/publication/272022137 Riemann Hypothesis Addendum - Breakthrough.}
\medskip
Here, we recall Albert Einstein's words on mathematics:
Blackholes are where God divided by zero.
I don't believe in mathematics.
George Gamow (1904-1968) Russian-born American nuclear physicist and cosmologist remarked that "it is well known to students of high school algebra" that division by zero is not valid; and Einstein admitted it as {\bf the biggest blunder of his life} [1]：
1. Gamow, G., My World Line (Viking, New York). p 44, 1970.
For our ideas on the division by zero, see the survey style announcements 179,185,237,246,247,250 and 252 of the Institute of Reproducing Kernels (\cite{ann179,ann185,ann237,ann246,ann247,ann250,ann252,ann293}).
\section{On mathematics}
Apparently, the division by zero is a great missing in our mathematics and the result (1.2) is definitely determined as our basic mathematics, as we see from Proposition 1. Note its very general assumptions and many fundamental evidences in our world in (\cite{kmsy,msy,mos}). The results will give great impacts on Euclidean spaces, analytic geometry, calculus, differential equations, complex analysis and physical problems. See our announcements for the details.
The mysterious history of the division by zero over one thousand years is a great shame of mathematicians and human race on the world history, like the Ptolemaic system (geocentric theory). The division by zero will become a typical symbol of foolish human race with long and unceasing struggles. Future people will realize this fact as a definite common sense.
We should check and fill our mathematics, globally and beautifully, from the viewpoint of the division by zero. Our mathematics will be more perfect and beautiful, and will give great impacts to our basic ideas on the universe.
\section{Albert Einstein's biggest blunder}
The division by zero is directly related to the Einstein's theory and various
physical problems
containing the division by zero. Now we should check the theory and the problems by the concept of the RIGHT and DEFINITE division by zero. Now is the best time since 100 years from Albert Einstein. It seems that the background knowledge is timely fruitful.
\section{Computer systems}
The above Professors listed are wishing the contributions in order to avoid the zero division trouble in computers. Now, we should arrange new computer systems in order not to meet the division by zero trouble in computer systems.
\section{General ideas on the universe}
The division by zero may be related to religion, philosophy and the ideas on the universe, and it will creat a new world. Look the new world.
\bigskip
We are standing on a new generation and in front of the new world, as in the discovery of the Americas.
\bigskip
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{bb}
J. P. Barukcic and I. Barukcic, Anti Aristotle—The Division of Zero by Zero. Journal of Applied Mathematics and Physics, {\bf 4}(2016), 749-761.
doi: 10.4236/jamp.2016.44085.
\bibitem{bht}
J. A. Bergstra, Y. Hirshfeld and J. V. Tucker,
Meadows and the equational specification of division (arXiv:0901.0823v1[math.RA] 7 Jan 2009).
\bibitem{cs}
L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.
\bibitem{kmsy}
M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,
New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,
Int. J. Appl. Math. {\bf 27} (2014), no 2, pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
\bibitem{ms}
T. Matsuura and S. Saitoh,
Matrices and division by zero $z/0=0$,
Linear Algebra \& Matrix Theory (ALAMT)(to appear).
\bibitem{msy}
H. Michiwaki, S. Saitoh, and M.Yamada,
Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math. {\bf 6}(2015), 1--8. http://www.ijapm.org/show-63-504-1.html
\bibitem{mos}
H. Michiwaki, H. Okumura, and S. Saitoh,
Division by Zero $z/0 = 0$ in Euclidean Spaces.
International Journal of Mathematics and Computation
(in press).
\bibitem{ra}
T. S. Reis and J.A.D.W. Anderson,
Transdifferential and Transintegral Calculus,
Proceedings of the World Congress on Engineering and Computer Science 2014 Vol I
WCECS 2014, 22-24 October, 2014, San Francisco, USA
\bibitem{ra2}
T. S. Reis and J.A.D.W. Anderson,
Transreal Calculus,
IAENG International J. of Applied Math., {\bf 45}(2015): IJAM 45 1 06.
\bibitem{s}
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. {\bf 4} (2014), no. 2, 87--95. http://www.scirp.org/journal/ALAMT/
\bibitem{ttk}
S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operations on the real and complex fields, Tokyo Journal of Mathematics, {\bf 38}(2015), no. 2, 369-380.
\bibitem{ann179}
Announcement 179 (2014.8.30): Division by zero is clear as z/0=0 and it is fundamental in mathematics.
\bibitem{ann185}
Announcement 185 (2014.10.22): The importance of the division by zero $z/0=0$.
\bibitem{ann237}
Announcement 237 (2015.6.18): A reality of the division by zero $z/0=0$ by geometrical optics.
\bibitem{ann246}
Announcement 246 (2015.9.17): An interpretation of the division by zero $1/0=0$ by the gradients of lines.
\bibitem{ann247}
Announcement 247 (2015.9.22): The gradient of y-axis is zero and $\tan (\pi/2) =0$ by the division by zero $1/0=0$.
\bibitem{ann250}
Announcement 250 (2015.10.20): What are numbers? - the Yamada field containing the division by zero $z/0=0$.
\bibitem{ann252}
Announcement 252 (2015.11.1): Circles and
curvature - an interpretation by Mr.
Hiroshi Michiwaki of the division by
zero $r/0 = 0$.
\bibitem{ann281}
Announcement 281(2016.2.1): The importance of the division by zero $z/0=0$.
\bibitem{ann282}
Announcement 282(2016.2.2): The Division by Zero $z/0=0$ on the Second Birthday.
\bibitem{ann293}
Announcement 293(2016.3.27): Parallel lines on the Euclidean plane from the viewpoint of division by zero 1/0=0.
\end{thebibliography}
\end{document}
2016-05-20 17:55:32 来源: 澎湃新闻网(上海)
（原标题：比尔·盖茨最新书单，有本书教你怎样用数学思维约到女神）
【编者按】
一如既往的，比尔·盖茨在初夏更新了个人博客文章，开出了一份推荐阅读书单。
比尔·盖茨最新书单：怎样用数学思维约到女神？
比尔·盖茨最新书单中涵盖了科学、数学和小说类别的5本书籍：美国科幻作家尼尔·斯蒂芬森（Neal Stephenson）的《七夏娃》（Seveneves）、英国生物化学家尼克·莱恩（Nick Lane）的《至关重要的问题》（The Vital Question）、日本电商大佬三木谷浩史与父亲三木谷良一合著的《竞争力》、以色列历史学教授尤瓦尔·赫拉利（Noah Yuval Harari）的《人类简史：从动物到上帝》（Sapiens：A Brief History of Humankind，中文版已出），以及威斯康星大学教授乔丹·爱伦伯格（Jordan Ellenberg）所著的这本How Not to be Wrong（中文本《魔鬼数学》，中信出版社，2015年9月）。
该书旨在说明数学是一门告诉我们“如果做才不会犯错”的科学。澎湃新闻“翻书党”经授权摘编其中关于如何运用数学思维让你成功约会心仪对象的章节。
比尔·盖茨最新书单：怎样用数学思维约到女神？
威斯康星大学教授乔丹·爱伦伯格（Jordan Ellenberg）
我们先从多头绒泡菌这个有趣的微生物说起，在大部分时间里，它都表现为一种极小的单细胞形态，即，单身，但是，如果条件合适，成千上万的多头绒泡菌就会结合在一起，形成“原质团”。在这种形态下，多头绒泡菌为嫩黄色，体积也会变大，肉眼可见。在野外环境中，它生长在腐朽的植物上，而在实验室中，它最喜欢的栖息之所是燕麦。
多头绒泡菌无脑无神经，更不用说情感与思维了，但是，与所有生物一样，它也会做决策，更有意思的是，多头绒泡菌会做出非常正确的决策。当然，决策无非是关于“靠近我喜欢的东西”（燕麦）与“远离我不喜欢的东西”（明亮的光线）。研究人员在皮式培养皿的一侧放置3 克燕麦，在另一侧放置5 克燕麦并用紫外线照射燕麦，然后在培养皿的中间位置放上多头绒泡菌。他们发现，在这种情况下，多头绒泡菌选择这两个方向的次数大约各占一半，更多的食物基本抵消了紫外线给多头绒泡菌造成的不舒服的感觉。若是经济学家来分析，他们肯定会认为，对于多头绒泡菌而言，黑暗中的一小堆燕麦与明亮处的一大堆燕麦的效用是一样的，因此，多头绒泡菌会左右为难。
不过，在把5 克燕麦换成10 克之后，这种平衡完全被打破了，多头绒泡菌根本不在乎光线的问题，每次都会朝10 克燕麦靠近。这个实验告诉我们，多头绒泡菌在做决策时会优先考虑哪些因素，在这些因素相互矛盾时又是如何做出选择的。从这些实验来看，多头绒泡菌似乎相当理性。
但是，当实验者把多头绒泡菌放到皮式培养皿中之后，给了它们三种选择：在黑暗处放置3 克燕麦（3– 黑暗），在明亮处放置5 克燕麦（5– 明亮），在黑暗处放置1 克燕麦（1– 黑暗）。我们可能会认为多头绒泡菌绝不可能靠近1– 黑暗，因为3– 黑暗的燕麦数量更多，具有明显的优势。的确，多头绒泡菌几乎一次也没有选择1– 黑暗。
我们还可能会进一步猜测，既然在之前的条件下，3– 黑暗与5– 明亮对多头绒泡菌具有同样的吸引力，那么，在新的条件下，应该会继续出现这样的情况。用经济学家的话来说，新的选择方案不会改变3– 黑暗与5– 明亮效用相同的事实。但是，实验结果并非如此：在有1– 黑暗可选的情况下，多头绒泡菌的喜好发生了变化，选择3– 黑暗的次数是5– 明亮的三倍！
这是怎么回事呢？
数学领域的一个流行术语—“无关选项的独立性”（independence of irrelevant alternatives），根据这个法则，无论你是多头绒泡菌、人还是民主国家，如果要在方案甲与方案乙之间做选择，第三个方案丙的出现都不会影响你对甲和乙的倾向性。如果你在为购买丰田普锐斯还是悍马犹豫不决，福特斑马对你到底买哪款车的选择不会产生任何影响，因为你知道自己肯定不会购买福特斑马。
人也是如此，我们把燕麦换成浪漫的伴侣。我们将为你提供几个虚构的人物，请把这几个人想象成你未来的约会对象，然后从这些人中选择一个与之约会。假定这几个未来的约会对象都满足以下条件：（1）北卡罗来纳大学（或者杜克大学）的学生；（2）与你同一个民族或种族；（3）与你年龄相当。我们会描述他们的几个特点，并就每种特点给出百分位数。这些百分位数反映了他们的某种特点在相同性别、种族与年龄的北卡罗来纳大学（或者杜克大学）学生中的相对位置。
亚当的魅力处于第81 百分位数，可信度处于第51 百分位数，智力处于第65 百分位数；
比尔的魅力处于第61 百分位数、可信度处于第51 百分位数，智力处于第87 百分位数。
与多头绒泡菌一样，这些实验对象面临着艰难的选择。他们给出的答案也与多头绒泡菌一样，选择亚当和比尔作为未来约会对象的大学生各占总数的一半。
但是，在克里斯出现之后，情况发生了变化。克里斯的魅力与可信度分别处于第81 百分位数和第51 百分位数，但是他的智力与亚当一样，处于第54 百分位数。克里斯是一个“无关选项”，因为他明显逊色于亚当和比尔。结果我们应该可以猜到：在稍有逊色的新版亚当出现之后，正版亚当似乎变得更有吸引力了。当面对亚当、比尔和克里斯这3 个选项时，接近2/3 的女性选择亚当作为约会对象。
所以，如果你是一位正在寻找真爱的单身汉，那么，在考虑与哪位朋友一起去城里赴心仪对象的约会时，应该选择条件与你相似但略微逊色于你的那位。
非理性从何而来呢？我们已经知道，完全理性的个体在集体行动中有可能扭曲真实的民意。但是经验告诉我们，个人不可能是完全理性的。关于多头绒泡菌的研究表明，我们的日常行为之所以自相矛盾或不一致，可能基于某种更彻底的理由。个人之所以不理性，也许是因为他们并不是真正的个体。每个人都是一个小国家，我们要做的就是尽可能地处理各种争端、做出妥协，而最后得到的未必都是合理的结果。就像多头绒泡菌一样，我们也有可能小错不断，但却能做到大错不犯。http://tech.163.com/16/0520/17/BNHDR6PG000915BF.html
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Announcement 300: New challenges on the division by zero z/0=0\\
(2016.05.22)}
\author{{\it Institute of Reproducing Kernels}\\
Kawauchi-cho, 5-1648-16,\\
Kiryu 376-0041, Japan\\
%\date{\today}
\maketitle
{\bf Abstract: } In this announcement, for its importance we would like to state the
situation on the division by zero and propose basic new challenges.
\bigskip
\section{Introduction}
%\label{sect1}
By a {\bf natural extension} of the fractions
\begin{equation}
\frac{b}{a}
\end{equation}
for any complex numbers $a$ and $b$, we found the simple and beautiful result, for any complex number $b$
\begin{equation}
\frac{b}{0}=0,
\end{equation}
incidentally in \cite{s} by the Tikhonov regularization for the Hadamard product inversions for matrices and we discussed their properties and gave several physical interpretations on the general fractions in \cite{kmsy} for the case of real numbers.
The division by zero has a long and mysterious story over the world (see, for example, Google site with the division by zero) with its physical viewpoints since the document of zero in India on AD 628, however,
Sin-Ei Takahasi (\cite{kmsy}) established a simple and decisive interpretation (1.2) by analyzing the extensions of fractions and by showing the complete characterization for the property (1.2):
\bigskip
{\bf Proposition 1. }{\it Let F be a function from ${\bf C }\times {\bf C }$ to ${\bf C }$ satisfying
$$
F (b, a)F (c, d)= F (bc, ad)
$$
for all
$$
a, b, c, d \in {\bf C }
$$
and
$$
F (b, a) = \frac {b}{a }, \quad a, b \in {\bf C }, a \ne 0.
$$
Then, we obtain, for any $b \in {\bf C } $
$$
F (b, 0) = 0.
$$
}
Note that the complete proof of this proposition is simply given by 2 or 3 lines.
\medskip
We thus should consider, for any complex number $b$, as (1.2);
that is, for the mapping
\begin{equation}
w = \frac{1}{z},
\end{equation}
the image of $z=0$ is $w=0$ ({\bf should be defined}). This fact seems to be a curious one in connection with our well-established popular image for the point at infinity on the Riemann sphere. Therefore, the division by zero will give great impacts to complex analysis and to our ideas for the space and universe.
However, the division by zero (1.2) is now clear, indeed, for the introduction of (1.2), we have several independent approaches as in:
\medskip
1) by the generalization of the fractions by the Tikhonov regularization or by the Moore-Penrose generalized inverse,
\medskip
2) by the intuitive meaning of the fractions (division) by H. Michiwaki,
\medskip
3) by the unique extension of the fractions by S. Takahasi, as in the above,
\medskip
4) by the extension of the fundamental function $W = 1/z$ from ${\bf C} \setminus \{0\}$ into ${\bf C}$ such that $W =1/z$ is a one to one and onto mapping from $ {\bf C} \setminus \{0\} $ onto ${\bf C} \setminus \{0\}$ and the division by zero $1/0=0$ is a one to one and onto mapping extension of the function $W =1/z $ from ${\bf C}$ onto ${\bf C}$,
\medskip
and
\medskip
5) by considering the values of functions with the mean values of functions.
\medskip
Furthermore, in (\cite{msy}) we gave the results in order to show the reality of the division by zero in our world:
\medskip
\medskip
A) a field structure containing the division by zero --- the Yamada field ${\bf Y}$,
\medskip
B) by the gradient of the $y$ axis on the $(x,y)$ plane --- $\tan \frac{\pi}{2} =0$,
\medskip
C) by the reflection $W =1/\overline{z}$ of $W= z$ with respect to the unit circle with center at the origin on the complex $z$ plane --- the reflection point of zero is zero,
\medskip
and
\medskip
D) by considering rotation of a right circular cone having some very interesting
phenomenon from some practical and physical problem.
\medskip
In (\cite{mos}), many division by zero results in Euclidean spaces are given and the basic idea at the point at infinity should be changed. In (\cite{ms}), we gave beautiful geometrical interpretations of determinants from the viewpoint of the division by zero. The results show that the division by zero is our basic and elementary mathematics in our world.
\medskip
See J. A. Bergstra, Y. Hirshfeld and J. V. Tucker \cite{bht} for the relationship between fields and the division by zero, and the importance of the division by zero for computer science. It seems that the relationship of the division by zero and field structures are abstract in their paper.
Meanwhile, J. P. Barukcic and I. Barukcic (\cite{bb}) discussed recently the relation between the divisions $0/0$, $1/0$ and special relative theory of Einstein. However, their logic seems to be curious and their results contradict with ours.
Furthermore, T. S. Reis and J.A.D.W. Anderson (\cite{ra,ra2}) extend the system of the real numbers by introducing an ideal number for the division by zero $0/0$.
Meanwhile, we should refer to up-to-date information:
{\it Riemann Hypothesis Addendum - Breakthrough
Kurt Arbenz
https://www.researchgate.net/publication/272022137 Riemann Hypothesis Addendum - Breakthrough.}
\medskip
Here, we recall Albert Einstein's words on mathematics:
Blackholes are where God divided by zero.
I don't believe in mathematics.
George Gamow (1904-1968) Russian-born American nuclear physicist and cosmologist remarked that "it is well known to students of high school algebra" that division by zero is not valid; and Einstein admitted it as {\bf the biggest blunder of his life} [1]：
1. Gamow, G., My World Line (Viking, New York). p 44, 1970.
For our ideas on the division by zero, see the survey style announcements 179,185,237,246,247,250 and 252 of the Institute of Reproducing Kernels (\cite{ann179,ann185,ann237,ann246,ann247,ann250,ann252,ann293}).
\section{On mathematics}
Apparently, the division by zero is a great missing in our mathematics and the result (1.2) is definitely determined as our basic mathematics, as we see from Proposition 1. Note its very general assumptions and many fundamental evidences in our world in (\cite{kmsy,msy,mos}). The results will give great impacts on Euclidean spaces, analytic geometry, calculus, differential equations, complex analysis and physical problems. See our announcements for the details.
The mysterious history of the division by zero over one thousand years is a great shame of mathematicians and human race on the world history, like the Ptolemaic system (geocentric theory). The division by zero will become a typical symbol of foolish human race with long and unceasing struggles. Future people will realize this fact as a definite common sense.
We should check and fill our mathematics, globally and beautifully, from the viewpoint of the division by zero. Our mathematics will be more perfect and beautiful, and will give great impacts to our basic ideas on the universe.
\section{Albert Einstein's biggest blunder}
The division by zero is directly related to the Einstein's theory and various
physical problems
containing the division by zero. Now we should check the theory and the problems by the concept of the RIGHT and DEFINITE division by zero. Now is the best time since 100 years from Albert Einstein. It seems that the background knowledge is timely fruitful.
\section{Computer systems}
The above Professors listed are wishing the contributions in order to avoid the zero division trouble in computers. Now, we should arrange new computer systems in order not to meet the division by zero trouble in computer systems.
\section{General ideas on the universe}
The division by zero may be related to religion, philosophy and the ideas on the universe, and it will creat a new world. Look the new world.
\bigskip
We are standing on a new generation and in front of the new world, as in the discovery of the Americas.
\bigskip
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{bb}
J. P. Barukcic and I. Barukcic, Anti Aristotle—The Division of Zero by Zero. Journal of Applied Mathematics and Physics, {\bf 4}(2016), 749-761.
doi: 10.4236/jamp.2016.44085.
\bibitem{bht}
J. A. Bergstra, Y. Hirshfeld and J. V. Tucker,
Meadows and the equational specification of division (arXiv:0901.0823v1[math.RA] 7 Jan 2009).
\bibitem{cs}
L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.
\bibitem{kmsy}
M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,
New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,
Int. J. Appl. Math. {\bf 27} (2014), no 2, pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
\bibitem{ms}
T. Matsuura and S. Saitoh,
Matrices and division by zero $z/0=0$,
Linear Algebra \& Matrix Theory (ALAMT)(to appear).
\bibitem{msy}
H. Michiwaki, S. Saitoh, and M.Yamada,
Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math. {\bf 6}(2015), 1--8. http://www.ijapm.org/show-63-504-1.html
\bibitem{mos}
H. Michiwaki, H. Okumura, and S. Saitoh,
Division by Zero $z/0 = 0$ in Euclidean Spaces.
International Journal of Mathematics and Computation
(in press).
\bibitem{ra}
T. S. Reis and J.A.D.W. Anderson,
Transdifferential and Transintegral Calculus,
Proceedings of the World Congress on Engineering and Computer Science 2014 Vol I
WCECS 2014, 22-24 October, 2014, San Francisco, USA
\bibitem{ra2}
T. S. Reis and J.A.D.W. Anderson,
Transreal Calculus,
IAENG International J. of Applied Math., {\bf 45}(2015): IJAM 45 1 06.
\bibitem{s}
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. {\bf 4} (2014), no. 2, 87--95. http://www.scirp.org/journal/ALAMT/
\bibitem{ttk}
S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operations on the real and complex fields, Tokyo Journal of Mathematics, {\bf 38}(2015), no. 2, 369-380.
\bibitem{ann179}
Announcement 179 (2014.8.30): Division by zero is clear as z/0=0 and it is fundamental in mathematics.
\bibitem{ann185}
Announcement 185 (2014.10.22): The importance of the division by zero $z/0=0$.
\bibitem{ann237}
Announcement 237 (2015.6.18): A reality of the division by zero $z/0=0$ by geometrical optics.
\bibitem{ann246}
Announcement 246 (2015.9.17): An interpretation of the division by zero $1/0=0$ by the gradients of lines.
\bibitem{ann247}
Announcement 247 (2015.9.22): The gradient of y-axis is zero and $\tan (\pi/2) =0$ by the division by zero $1/0=0$.
\bibitem{ann250}
Announcement 250 (2015.10.20): What are numbers? - the Yamada field containing the division by zero $z/0=0$.
\bibitem{ann252}
Announcement 252 (2015.11.1): Circles and
curvature - an interpretation by Mr.
Hiroshi Michiwaki of the division by
zero $r/0 = 0$.
\bibitem{ann281}
Announcement 281(2016.2.1): The importance of the division by zero $z/0=0$.
\bibitem{ann282}
Announcement 282(2016.2.2): The Division by Zero $z/0=0$ on the Second Birthday.
\bibitem{ann293}
Announcement 293(2016.3.27): Parallel lines on the Euclidean plane from the viewpoint of division by zero 1/0=0.
\end{thebibliography}
\end{document}
0 件のコメント:
コメントを投稿